Chapter 8 – Theoretical Biophysics 367
b The three different regions of motor behavior are equivalent to the “binding
zone” (x0 < x < x0 + Δx), the “bound zone” (x0 < x < 0), and the “unbinding zone”
(x > 0). For the binding zone, kon is constant, koff is zero, and the initial boundary
condition at t = 0 is P = 0. Solving Equation 8.99 under these conditions
indicates a solution of
(8.102)
P
k
x
x
v
= −
−
−
(
)
1
0
exp
on
In the bound zone, the probability of binding will be a constant Pc and will match
that in the binding zone when x = x0:
(8.103)
P
P
k
x
v
b
c
on
exp
=
= −
−
1
∆
In the unbinding zone, kon is zero, koff is a constant, and the initial boundary condi
tion at x = 0 is P = Pb. Thus, solving Equation 8.99 under these conditions indicates
(8.104)
P
C
k
x
v
b
off
exp
=
−
Thus, for a motor–track interaction with the same binding/unbinding kinetics,
a “slow” motor indicates a steeper magnitude of gradient of P with respect to
x compared to a “fast” motor at equivalent x and has a larger value of Pc (see
Figure 8.7b).
c The mean work done per binding site repeat across which the motor translocates
is given by
(8.105)
W
F x P x dx
xP dx
xP
k
x
v
c
x
c
off
=
( ) ( )
=
−(
)
+
−
−
∫
∫
1bindingsite
κ
κ
0
0
exp
∞
∫
0
dx
The first term is trivial; the second term can be solved by parts:
(8.106)
W
P x
P v
k
k
x
v
x
v
k
c
c
off
on
off
=
−
−
−
−
1
2
1
2
0
2
2
2
0
2
2
2
κ
κ
κ
exp
∆
Thus, the average force per binding site repeat is
(8.107)
〈〉=
=
−
−
−
F
W
d
d
k
x
v
x
v
k
κ 1
2
0
2
2
2
exp
on
off
∆
A sketch of the variation of〈F〉with v is shown in Figure 8.7c.